Exponent rules and polynomials

Product rule

Same base, multiplying: add the exponents. Multiply the coefficients normally.

Example: \(x^3 \cdot x^5 = x^8\); \(\;(3x^2y)(4x^5y^3) = 12x^7y^4\).

Watch out: \(2^3 \cdot 2^4 = 2^7\), not \(4^7\). The base stays the same.

Quotient rule

Same base, dividing: subtract the exponents (top minus bottom).

Example: \(\dfrac{x^9}{x^4} = x^5\); \(\;\dfrac{12x^5y^2}{4x^2y} = 3x^3y\).

Power rules

A power of a power: multiply the exponents. A power of a product or quotient: give the exponent to every factor.

Example: \((x^4)^3 = x^{12}\); \(\;(2x^3)^4 = 16x^{12}\); \(\;\left(\dfrac{x^2}{3}\right)^3 = \dfrac{x^6}{27}\).

Watch out: \((-3x)^2 = 9x^2\): the \(-3\) is squared too.

Zero and negative exponents

\(a^0 = 1\). A negative exponent means "move it across the fraction bar and make the exponent positive". \(\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n\).

Example: \(5x^0 = 5\) but \((5x)^0 = 1\); \(\;2^{-3} = \dfrac{1}{8}\); \(\;\dfrac{x^{-2}}{y^{-3}} = \dfrac{y^3}{x^2}\); \(\;\left(\dfrac{2}{3}\right)^{-2} = \dfrac{9}{4}\).

Watch out: a negative exponent does not make the number negative: \(2^{-3}\) is positive.

Putting the rules together

Work inside parentheses first, then powers, then combine like bases. Finish with positive exponents only.

Example: \(\dfrac{(2x^3y^{-1})^2}{4x^{-2}y} = \dfrac{4x^6y^{-2}}{4x^{-2}y} = x^8y^{-3} = \dfrac{x^8}{y^3}\).

Scientific notation

Scientific notation: \(a \times 10^n\) with \(1 \le |a| \lt 10\). Big numbers get positive \(n\), small numbers negative \(n\). To multiply or divide: handle the numbers and the powers of 10 separately, then fix \(a\).

Example: \(45{,}000 = 4.5 \times 10^4\); \(\;0.0032 = 3.2 \times 10^{-3}\).

Example: \((3 \times 10^4)(4 \times 10^{-7}) = 12 \times 10^{-3} = 1.2 \times 10^{-2}\); \(\;\dfrac{6 \times 10^5}{2 \times 10^{-2}} = 3 \times 10^7\).

Polynomial words

A polynomial is a sum of terms with whole-number exponents. Monomial: 1 term. Binomial: 2. Trinomial: 3. The degree is the highest exponent.

Example: \(4x^3 - x + 7\) is a trinomial of degree 3; its leading coefficient is 4.

Adding and subtracting polynomials

Combine like terms. To subtract, change the sign of every term in the second polynomial first.

Example: \((5x^2 - 3x + 2) - (2x^2 + x - 6) = 5x^2 - 3x + 2 - 2x^2 - x + 6 = 3x^2 - 4x + 8\).

Multiplying polynomials

Multiply every term of the first by every term of the second (FOIL or a box for two binomials), then combine like terms.

Example: \((2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12\).

Example: \((x + 2)(x^2 - 3x + 1) = x^3 - 3x^2 + x + 2x^2 - 6x + 2 = x^3 - x^2 - 5x + 2\).

Special products

\((a + b)^2 = a^2 + 2ab + b^2\); \(\;(a - b)^2 = a^2 - 2ab + b^2\); \(\;(a + b)(a - b) = a^2 - b^2\).

Example: \((x + 3)^2 = x^2 + 6x + 9\); \(\;(2x - 5)^2 = 4x^2 - 20x + 25\); \(\;(3x + 4)(3x - 4) = 9x^2 - 16\).

Watch out: \((x + 3)^2 \ne x^2 + 9\). Do not forget the middle term \(2ab\).

Dividing by a monomial

Divide every term of the top by the bottom.

Example: \(\dfrac{6x^3 - 9x^2 + 3x}{3x} = 2x^2 - 3x + 1\).

Long division by a binomial

Divide the first terms, multiply, subtract, bring down, repeat. Write missing powers with 0 coefficients. Answer: quotient \(+ \dfrac{\text{remainder}}{\text{divisor}}\). Check: quotient × divisor + remainder = the original.

Example: \((x^2 + 5x + 7) \div (x + 2) = x + 3 + \dfrac{1}{x + 2}\).

Example: \((x^3 - 8) \div (x - 2)\): write \(x^3 + 0x^2 + 0x - 8\); the quotient is \(x^2 + 2x + 4\), remainder 0.

Shortcut: synthetic division

Optional, only for dividing by \(x - c\). Write \(c\) and the coefficients; bring down, multiply by \(c\), add, repeat. The last number is the remainder.

Example: \((x^3 - 8) \div (x - 2)\): use 2 with 1, 0, 0, −8. The bottom row is 1, 2, 4, 0, so \(x^2 + 2x + 4\), remainder 0.